1. Technical Field
The present invention relates to risk model aggregation. More particularly, the present invention relates to a method and apparatus for a mathematical technique that forces an aggregate risk model to be consistent with embedded standalone models.
2. Description of the Prior Art    My ventures are not in one bottom trusted,    Nor to one place; nor is my whole estate    Upon the fortune of this present year.    Therefore, my merchandise makes me not sad.            William Shakespeare, The Merchant of Venice (1598)        
Shakespeare reminds us that the perils of investment have been with us always. Investors have long appreciated the importance of diversification. The idea took quantitative form with the work of H. Markowitz, The Birth of Mean-Variance Optimization (7.1, 77-91), Portfolio Selection, Journal of Finance (1952). Since then increasingly sophisticated mathematical and statistical tools have been brought to bear on the problem of estimating the aggregate risk of a portfolio.
The aggregate risk of a portfolio depends crucially on the covariances of the portfolio's constituent assets. Unfortunately, in practical situations so many covariances come into play that it is impossible to estimate all of them directly from historical data. Factor models overcome this difficulty by expressing the large number of asset covariances in terms of a small number of factor covariances.
A factor model is defined through a linear regression, as follows:y=A·x+ε.  (1)
Here y is the vector of asset returns whose variances and covariances require estimation, x is a vector of factor returns whose variances and covariances can be reliably estimated, and A is a matrix, specified a priori, that describes the sensitivities of the assets to the factors. The vector ε of errors is usually assumed to be normally distributed with a diagonal covariance matrix D. This model estimates the asset covariance matrix, Σ(y), asΣ(y)≈A·Σ(x)·AT+D  (2)where Σ(x) is the matrix of factor covariances. The dimension of the matrix Σ(x) is small enough that historical data allow reasonable estimates of the covariances.
As quantitative risk management has become more sophisticated, factor models have become more finely detailed. At the same time, models have broadened; large models encompassing many asset classes and markets have become necessary for large firms to forecast their “total risk.” Both of these trends have forced the number of factors upward to the point where, once again, more covariances are required (this time, between factors instead of assets) than can be accurately estimated.
This problem can be addressed by iterating the idea that worked before. Factor models themselves can be built up from smaller factor models. However, this approach introduces a new set of difficulties related to consistency. To illustrate the problem, imagine building a factor model to estimate risk for a portfolio composed of US equity and fixed income securities. Suppose further that there are already excellent standalone factor models that separately treat equities and fixed income securities. An aggregate factor model will almost certainly be inconsistent with the standalone factor models. The discrepancies might result from the iterated factor structure, from differences in the amount and frequency of data, from clashes in statistical methods specific to each standalone model, or from other sources. Such discrepancies are undesirable in part because they can cause different levels of a firm to have different views of the same source of investment risk.
It would be advantageous to provide a mathematical technique that enforces consistency between an aggregate model and the standalone models, i.e., to achieve breadth without sacrificing meaningful detail. More specifically, it would be advantageous to revise the aggregate risk model to be consistent with the standalone models. Unfortunately, enforcing consistency almost always involves the destruction of covariance Information in the aggregate model. The problem is to minimize the damage.